Ultragraph C*-algebra

In mathematics, an ultragraph C*-algebra is a universal C*-algebra generated by partial isometries on a collection of Hilbert spaces constructed from ultragraphs.^[1]^{pp. 6-7}. These C*-algebras were created in order to simultaneously generalize the classes of graph C*-algebras and Exel–Laca algebras, giving a unified framework for studying these objects.^[1] This is because every graph can be encoded as an ultragraph, and similarly, every infinite graph giving an Exel-Laca algebras can also be encoded as an ultragraph.

Definitions

Ultragraphs

An ultragraph ${\mathcal {G}}=(G^{0},{\mathcal {G}}^{1},r,s)$ consists of a set of vertices $G^{0}$ , a set of edges ${\mathcal {G}}^{1}$ , a source map $s:{\mathcal {G}}^{1}\to G^{0}$ , and a range map $r:{\mathcal {G}}^{1}\to P(G^{0})\setminus \{\emptyset \}$ taking values in the power set collection $P(G^{0})\setminus \{\emptyset \}$ of nonempty subsets of the vertex set. A directed graph is the special case of an ultragraph in which the range of each edge is a singleton, and ultragraphs may be thought of as generalized directed graph in which each edges starts at a single vertex and points to a nonempty subset of vertices.

Example

Ultragraph visualization — Ultragraph $(\{v,w,x\},\{e,f,g\},s,r)$

An easy way to visualize an ultragraph is to consider a directed graph with a set of labelled vertices, where each label corresponds to a subset in the image of an element of the range map. For example, given an ultragraph with vertices and edge labels

${\mathcal {G}}^{0}=\{v,w,x\}$ , ${\mathcal {G}}^{1}=\{e,f,g\}$

with source an range maps

${\begin{matrix}s(e)=v&s(f)=w&s(g)=x\\r(e)=\{v,w,x\}&r(f)=\{x\}&r(g)=\{v,w\}\end{matrix}}$

can be visualized as the image on the right.

Ultragraph algebras

Given an ultragraph ${\mathcal {G}}=(G^{0},{\mathcal {G}}^{1},r,s)$ , we define ${\mathcal {G}}^{0}$ to be the smallest subset of $P(G^{0})$ containing the singleton sets $\{\{v\}:v\in G^{0}\}$ , containing the range sets $\{r(e):e\in {\mathcal {G}}^{1}\}$ , and closed under intersections, unions, and relative complements. A Cuntz–Krieger ${\mathcal {G}}$ -family is a collection of projections $\{p_{A}:A\in {\mathcal {G}}^{0}\}$ together with a collection of partial isometries $\{s_{e}:e\in {\mathcal {G}}^{1}\}$ with mutually orthogonal ranges satisfying

$p_{\emptyset }$ , $p_{A}p_{B}=p_{A\cap B}$ , $p_{A}+p_{B}-p_{A\cap B}=p_{A\cup B}$ for all $A\in {\mathcal {G}}^{0}$ ,
$s_{e}^{*}s_{e}=p_{r(e)}$ for all $e\in {\mathcal {G}}^{1}$ ,
$p_{v}=\sum _{s(e)=v}s_{e}s_{e}^{*}$ whenever $v\in G^{0}$ is a vertex that emits a finite number of edges, and
$s_{e}s_{e}^{*}\leq p_{s(e)}$ for all $e\in {\mathcal {G}}^{1}$ .

The ultragraph C*-algebra $C^{*}({\mathcal {G}})$ is the universal C*-algebra generated by a Cuntz–Krieger ${\mathcal {G}}$ -family.

Properties

Every graph C*-algebra is seen to be an ultragraph algebra by simply considering the graph as a special case of an ultragraph, and realizing that ${\mathcal {G}}^{0}$ is the collection of all finite subsets of $G^{0}$ and $p_{A}=\sum _{v\in A}p_{v}$ for each $A\in {\mathcal {G}}^{0}$ . Every Exel–Laca algebras is also an ultragraph C*-algebra: If $A$ is an infinite square matrix with index set $I$ and entries in $\{0,1\}$ , one can define an ultragraph by $G^{0}:=$ , $G^{1}:=I$ , $s(i)=i$ , and $r(i)=\{j\in I:A(i,j)=1\}$ . It can be shown that $C^{*}({\mathcal {G}})$ is isomorphic to the Exel–Laca algebra ${\mathcal {O}}_{A}$ .^[1]

Ultragraph C*-algebras are useful tools for studying both graph C*-algebras and Exel–Laca algebras. Among other benefits, modeling an Exel–Laca algebra as ultragraph C*-algebra allows one to use the ultragraph as a tool to study the associated C*-algebras, thereby providing the option to use graph-theoretic techniques, rather than matrix techniques, when studying the Exel–Laca algebra. Ultragraph C*-algebras have been used to show that every simple AF-algebra is isomorphic to either a graph C*-algebra or an Exel–Laca algebra.^[2] They have also been used to prove that every AF-algebra with no (nonzero) finite-dimensional quotient is isomorphic to an Exel–Laca algebra.^[2]

While the classes of graph C*-algebras, Exel–Laca algebras, and ultragraph C*-algebras each contain C*-algebras not isomorphic to any C*-algebra in the other two classes, the three classes have been shown to coincide up to Morita equivalence.^[3]

Notes

^ ^a ^b ^c A unified approach to Exel–Laca algebras and C*-algebras associated to graphs, Mark Tomforde, J. Operator Theory 50 (2003), no. 2, 345–368.
^ ^a ^b Realization of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras, Takeshi Katsura, Aidan Sims, and Mark Tomforde, J. Funct. Anal. 257 (2009), no. 5, 1589–1620.
^ Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence, Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde, J. Reine Angew. Math. 640 (2010), 135–165.

[Tom-1] A unified approach to Exel–Laca algebras and C*-algebras associated to graphs, Mark Tomforde, J. Operator Theory 50 (2003), no. 2, 345–368.

[KST-2] Realization of AF-algebras as graph algebras, Exel–Laca algebras, and ultragraph algebras, Takeshi Katsura, Aidan Sims, and Mark Tomforde, J. Funct. Anal. 257 (2009), no. 5, 1589–1620.

[3] Graph algebras, Exel–Laca algebras, and ultragraph algebras coincide up to Morita equivalence, Takeshi Katsura, Paul Muhly, Aidan Sims, and Mark Tomforde, J. Reine Angew. Math. 640 (2010), 135–165.

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